My wife has dual citizenship and for her it is a little more complicated since her passports do not have the same name (doing a name change on her Russian passport is a long bureaucratic process involving getting a legal name change). She has no issues when traveling, she exits and enters the US and most other countries using her US passport, but of course she has to show her Russian passport to board any flights to Russia, and of course when entering and exiting Russia. Sometimes when she transits a country that only allows a few days visa free transit (I.e. China), she has to show her Russian passport and onward flight information when leaving the US to prove she doesn’t need a visa since her return flight (also a transit) is beyond the visa free period. Several times when entering China, one of their boarder agents insists she cannot enter with no visa, but my wife persists, shows her onward flight, etc.
My whole point here is to say travel as a dual citizen can be complicated, but it is really no big deal. My wife has never had any denials, just sometimes some brief delays.